Actually, I tried 4x today, and every single times the P/S worked fine for about a minute while driving, and no more after that until I switched the car off and on again.
I am guessing that my P/S relay was misbehaving and the replacement one we put in works fine but we didn't realized it because after a minute the P/S stops working anyway due to the missing position sensor (or troublesome P/S ECU?)?

the steering sensor part number is 84305-17010
 
http://www.toyodiy.com/parts/p_J_199107_TOYOTA_MR2_SW20-ACMQF_8401.5.html
 
the sensor unscrews from the indicator stalk and has an electrical plug going to it.
i came across one were it was taken off and tucked into the dash to make the (wrong)boss kit fit
and also because (the owner) saw the wires on it and wasnt sure about disconnecting it!

Cheers mate, I was looking for the part number. I'll probably look for a used one...

Just a quick update, we replaced the P/S ECU and I tried with 2 different p/s relay (including bypassing the relay completely) and that still didn't fix it. Like Timbo said before, he tried to diagnose the steering sensor, and it looked like it was working fine.
I have absolutely no idea what else could be causing this.

About the indicator stalk.
I have the same problem and Terry at Road and Track told me that you have to change the assembly to fix this.
It looks like a common problem

Any more input from anyone?
Steering sensor seems fine and no code stored in there; steering fluid is good; P/S pump seems functional; P/S ECU and relay have been replaced (also tested with the relay bypassed). The P/S steering always works for 60 seconds and nothing more. I have noticed a few times that after going over a speed bump, the P/S would seem to work again for a few seconds and then the steering becomes quite 'heavy' again.
I am wondering if maybe a previous owner, at some point, put some kind of stand-alone or piggy-back ECU and would have messed up the P/S somehow, not sure if that makes any sense though? I really can't think of anything else.
 
If I can't find a way to fix it, maybe I'll just remove the parts from the car to save some weight and sell them, since they are not doing any good in there. In the meantime though, any help appreciated :)